A096000 Cupolar numbers: a(n) = (n+1)*(5*n^2 + 7*n + 3)/3.

1, 10, 37, 92, 185, 326, 525, 792, 1137, 1570, 2101, 2740, 3497, 4382, 5405, 6576, 7905, 9402, 11077, 12940, 15001, 17270, 19757, 22472, 25425, 28626, 32085, 35812, 39817, 44110, 48701, 53600, 58817, 64362, 70245, 76476, 83065, 90022, 97357, 105080, 113201, 121730

Offset: 0

Author

N. J. A. Sloane, in memory of Harold Scott MacDonald Coxeter [Feb 09 1907 - Mar 31 2003], May 08 2004

Keywords

• nonn
• easy

Comments

Number of equal balls that will fill a triangular cupola, formed by splitting a cuboctahedron along one of its four "equilateral" hexagons.

Also as a(n) = (1/6)*(10*n^3 - 6*n^2 + 10*n), n>0: structured pentagonal anti-prism numbers (Cf. A100185 = structured anti-prisms); and structured tetragonal anti-diamond numbers (vertex structure 7) (Cf. A000447 = alternate vertex; A100188 = structured anti-diamonds). Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

References

• H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000447, A005902, A100145, A100185, A100188.

Programs

Magma

[(n+1)*(5*n^2+7*n+3)/3 : n in [0..50]]; // Wesley Ivan Hurt, May 23 2015

Maple

A096000:=n->(n+1)*(5*n^2+7*n+3)/3; seq(A096000(n), n=0..50); # Wesley Ivan Hurt, Mar 11 2014

Mathematica

Table[(n + 1)(5n^2 + 7n + 3)/3, {n, 0, 50}] (* Wesley Ivan Hurt, Mar 11 2014 *)
CoefficientList[Series[(1 + 6 x + 3 x^2)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 23 2015 *)

PARI

a(n) = (1/3)*(n+1)*(5*n^2+7*n+3) \\ Michel Marcus, Jul 11 2013

Formula

a(n) = (1/2)*(Q(n) + 3*n^2 + 3*n + 1), where Q(n) are the cuboctahedral numbers, A005902.

G.f.: (1+6*x+3*x^2)/(1-x)^4. - Paul Barry, Oct 28 2006

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Wesley Ivan Hurt, May 23 2015

E.g.f.: exp(x)*(3 + 27*x + 27*x^2 + 5*x^3)/3. - Elmo R. Oliveira, Aug 11 2025